Thunderbird only supports moving from one mailbox to another, it doesn't support syncing whole namespaces. It also doesn't support syncing with namespace rewriting or syncing flags, timestamps or access rights. It does not allow using different authorization and authentication and is unsuitable for large syncs as you can't resume anything or have a progress bar. It is also pretty slow, because it does one mail at a time and is just not designed for this.

Note that I used the proper IMAP terms. A mailbox is what the layman probably calls a folder or directory, a namespace is what the laymen probably would call a mailbox or account.